The automorphism group of a free-by-cyclic group in rank 2

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Polynomial-time complexity for instances of the endomorphism problem in free groups

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F, there is an endomorphism Φ of F sending W to U. This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a two- generator word. We show that when W is a two-generator word this algorithm solves the problem in time polynomial in the length of U. This result gives a polynomial-time algorithm for solving, in free groups, two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side.

Distortion of surface groups in Cat (0) free-by-cyclic groups

Given a non-positively curved 2-complex with a circle-valued Morse function satisfying some extra combinatorial conditions, we describe how to locally isometrically embed this in a larger non- positively curved 2-complex with free-by-cyclic fundamental group. This embedding procedure is used to produce examples of CAT(0) free-by-cyclic groups that contain closed hyperbolic surface subgroups with polynomial distortion of arbitrary degree. We also produce examples of CAT(0) hyperbolic free-by-cyclic groups that contain closed hyperbolic surface subgroups that are exponentially distorted.

Free and fragmenting filling length

The filling length of an edge-circuit η in the Cayley 2-complex of a finite presentation of a group is the minimal integer length L such that there is a combinatorial null-homotopy of η down to a base point through loops of length at most L. We introduce similar notions in which the full-homotopy is not required to fix a base point, and in which the contracting loop is allowed to bifurcate. We exhibit a group in which the resulting filling invariants exhibit dramatically different behaviour to the standard notion of filling length. We also define the corresponding filling invariants for Riemannian manifolds and translate our results to this setting.

Anfíbios Anuros da coleção Adolpho Lutz: III - Hyla claresignata Lutz & B. Lutz, 1939

Hyla claresignata Lutz & Lutz, 1939, is a large species apparently not closely allied to the other known Brazilian hylas. It is characterized by the very small tympanum; the head is short and the snout rounded; the legs are long, the hands and feet unusually large, the latter extensively webbbed. The specific name is derived from the insular, irregular, or roughly triangular, dark spots, with a light halo, found mostly in the dorso-lateral region and on the legs. It belongs to the rain-forest fauna of the Marítime Range. The adult is a bromeliad-dweller and the tadpole rhyacophilous. DESCRIPTION. Vomerine teeth in two separate, oblique, groups, behind the large choanae, parallel to the posterior half of their inner border. Tongue entire, short, very broad and hardly free behind. Snout short, rounded, with distinct canthus rostralis and gradually sloping loreal region. Eye very large and prominent, its horizontal diameter almost equal to the distance between its anterior corner and the tip of the snout. Tympanum very small, less than one third of the diameter of the eye, but distinct, partly covered by a short, heavy ridge. Lateral fingers less than one third webbed; fourth finger slightly longer than the second, just reaching the base of the disk of the third; subarticular tubercles well developed; an angular pollex rudiment, more noticeable in the males. Toes almost completely webbed, the edge of the web inserted at the base of the disk on the third and the fifth; an inner metatarsal tubercle. Skin smooth above, granular beneath, on the throat minutely so. No dermal appendage on the hell. Habit robust, head broader than long, body rather heavy, slightly narrowed in the postaxillary region. Legs long, the tibiotarsal articulation reaching beyond the tip of the snout when adpressed. Type (female): 61 mm. (Fig. 1.) DIAGNOSIS of TADPOLE (by G. Orton). "A large specialized, mountain-stream tadpole, with wide head an elongated, flattened snout, greatly enlarged lips and high tooth formula. Eyes dorsal. Spiracle sinistral, projecting, situated far back on side. Anus dextral. Tooth formula 8/12 to 9/14 in fully grown larvae. Tail with a prominent, vertical dark band across musculature and fins; a second concentration of dark pigment near tip of tail, may or may not form a similar but narrower band. Maximum known total length: 60mm.; head and body length 25mm. (Figs. 6 e 7). For further details see Lutz & Lutz, 1939 and Lutz B. & Orton G. 1946.

Free triples, large indifference classes and the majority rule

We present a new domain of preferences under which the majority relation is always quasi-transitive and thus Condorcet winners always exist. We model situations where a set of individuals must choose one individual in the group. Agents are connected through some relationship that can be interpreted as expressing neighborhood, and which is formalized by a graph. Our restriction on preferences is as follows: each agent can freely rank his immediate neighbors, but then he is indifferent between each neighbor and all other agents that this neighbor "leads to". Hence, agents can be highly perceptive regarding their neighbors, while being insensitive to the differences between these and other agents which are further removed from them. We show quasi-transitivity of the majority relation when the graph expressing the neighborhood relation is a tree. We also discuss a further restriction allowing to extend the result for more general graphs. Finally, we compare the proposed restriction with others in the literature, to conclude that it is independent of any previously discussed domain restriction.

Morfologia do Mycobacterium leprae hominis e do M. leprae muris: estudo baseado na microscopia electrônica e de contraste de fases

Hansen's Bacillus: By electron microscopy this bacillus shows membrane and halo, this being more visible when sorrounding the globi or bundles of bacilli; shows, also, free granules of various sizes which were before considered as dust of the dyes; shows external granules bound with the membrane and some times branching. By phases contrast microscopy examining leproma suspensions and subcataneous lymph at 400 x we saw many free granules with intense rotatory movement; granulated bacilli with screw, skip or stroke motion, producing slow progressive motion. All such elementes are surrounded by a halo, corresponding to the classical gloea. By a patient and delayed examination we were able to see that the internal granules are motile and help the progression of the bacilli, giving the impression that the cytoplasm is liquid. By a lasting observation we could see the larger granules form prolapse, like a pseudopode and abandon the bacilli and going in very rapid rotatory movement. There are branched bacilli; there are pedunculated fred granules like comets. The addition of a drop of formol at the preparation stops all movements. Stefansky's Bacillus: Repeated examination by RCA electron microscope, type EMU-25 of fresh suspensions of rat lepromas, led us to confirm the close relationship between human and murine leprosy agents. We examined also material from carabo (Lepra bubalorum) from Java, but due to fixation, the material was unsuitable for comparative studies. The Stefansky's bacilli showed also emmbranes and halos, internal or external granules (smaller than those of Hansen's bacillus). The bacilli shaded by chromium look thicker and shorter than those of Hansen. Due to electron bombardment both, Hansen's and Stefansky's baccilli suffer considerable alterations in their structure, showing black barrs of chromatin condensation at their extremities as also in their centers. By phase microscopy the Stefansky's bacilli showed elements with 1, 2 (bipolar), 3 or more internal small granules, developing identical movements as those of Hansen. The globi seem to be non-motile but the free bacilli appearing around the globi show intense movement. At 1000 x the examination is less satisfactory than at 400 x. The addition of formol solution in the preparation suppresses all movements, even the brownian, but the material becomes more suitable for the study of static morphology of the bacilli. CONCLUSION - The electron and phases contrast microscopy of leprous material from different types and phases of the disease may explain some of the unknown aspects of the biology and morphology of the bacilli.

Ecological sustainability and urban form

One controversial idea present in the debate on urban sustainability is that urban sprawl is an ecological stressing problem. We have tested this popular assumption by measuring the ecological footprint of commuting and housing of the 163 municipalities of the Barcelona Metropolitan Region and by relating the estimated values with residential density and accessibility, the fundamental determinant of residential density according to the Monocentric City Model.

Cat(0) and Cat (-1) dimensions of torsion free

We show that a particular free-by-cyclic group has CAT(0) dimension equal to 2, but CAT(-1) dimension equal to 3. We also classify the minimal proper 2-dimensional CAT(0) actions of this group; they correspond, up to scaling, to a 1-parameter family of locally CAT(0) piecewise Euclidean metrics on a fixed presentation complex for the group. This information is used to produce an infinite family of 2-dimensional hyperbolic groups, which do not act properly by isometries on any proper CAT(0) metric space of dimension 2. This family includes a free-by-cyclic group with free kernel of rank 6.

Algebraic extensions in free groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical fieldt heoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.

The density of injective endomorphisms of a free group

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Repeated games played in a network

Delayed perfect monitoring in an infinitely repeated discounted game is modelled by letting the players form a connected and undirected network. Players observe their immediate neighbors' behavior only, but communicate over time the repeated game's history truthfully throughout the network. The Folk Theorem extends to this setup, although for a range of discount factors strictly below 1, the set of sequential equilibria and the corresponding payoff set may be reduced. A general class of games is analyzed without imposing restrictions on the dimensionality of the payoff space. This and the bilateral communication structure allow for limited results under strategic communication only. As a by-product this model produces a network result; namely, the level of cooperation in this setup depends on the network's diameter, and not on its clustering coefficient as in other models.

Form, entwicklung und funktion der hypodermalen druesen von rhinocricus padbergii (Diplopoda)

Es werden die hypodermalen Druesen von rhinocricus padbergii (Diplopoda) histologisch beschrieben, sowie ihre Bildung in der Hypodermis sich frisch haeutender Tiere dargestellt. 1. Die Druesen bestehen aus vier wohldifferenzierten Zellen, von denen zwei als Kanalzellen und zwei als Druesenzellen funktionieren. 2. Die Druesenzellen entleeren ihr Sekret durch einen langen, sehr duennen Kanal auf die Oberflaeche der Cuticula im Anfang ihrer Neubildung, wo das Sekret hoechst wahrscheinlich zur Bildung der neuen Epicuticula beitraegt. 3. Der Ausleitungsapparat besteht aus einem in den beiden kanalzellen durch zwei aeussere Spiralfaeden verstaerkten, sehr duennen Kanal, der an der Basalflaeche der zweiten Kanalzelle eine klammerartige Verdickung zeigt. Er setzt sich als sehr duennhaeutiger, dehnbarer kanal in die erste Druesenzelle fort, an deren Basisflaeche er sich in zwei kurze Zweige aufteilt, von denen einer in die zweite Druesenzelle reicht. Um die offene Endigung der Zweige bildet sich eine kugelfoermige, radialstrahlige Ausfuehrzone, die vermutlich aus Mikrovilli aufgebaut ist. 4. Das Sekret kann in dem dehnbaren Teil des kanals innerhalb der ersten Druesenzelle gespeichert werden. Sein Austritt wird durch die Elastizitaet der erwaehnten Klammer des Kanals geregelt. 5. Die vier Kerne des Druesenkomplexes bilden sich durch zweifache amitotische Teilung aus einer Hypodermiszelle, besonders an den hinteren Raendern der Sklerite. 6. Der Ausfuehrkanal bildet sich als Invagination der aeussersten Ektocuticularlamelle.

On the number of K3,3-minor-free and maximal K3,3-minor-free graphs

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Strongly non-degenerate Lie algebras

Let A be a semiprime 2 and 3-torsion free non-commutative associative algebra. We show that the Lie algebra Der(A) of(associative) derivations of A is strongly non-degenerate, which is a strong form of semiprimeness for Lie algebras, under some additional restrictions on the center of A. This result follows from a description of the quadratic annihilator of a general Lie algebra inside appropriate Lie overalgebras. Similar results are obtained for an associative algebra A with involution and the Lie algebra SDer(A) of involution preserving derivations of A